Almost All Words Are Seen In Critical Site Percolation On The Triangular Lattice
Harry Kesten, Cornell University
Vladas Sidoravicius, IMPA
Yu Zhang, University of Colorado
Abstract
We consider critical site percolation on the triangular lattice, that
is, we choose $X(v) = 0$ or 1 with probability 1/2 each, independently
for all vertices $v$ of the triangular lattice. We say that a word
$(xi_1, xi_2,dots) in {0,1}^{Bbb N}$ is seen in the
percolation configuration if there exists a selfavoiding path $(v_1,
v_2, dots)$ on the triangular lattice with $X(v_i) = xi_i, i ge
1$. We prove that with probability 1 "almost all" words, as well as
all periodic words, except the two words $(1,1,1, dots)$ and
$(0,0,0,dots)$, are seen. "Almost all" words here means almost all
with respect to the measure $mu_beta$ under which the $xi_i$ are
i.i.d. with $mu_beta {xi_i = 0}=1 - mu_beta {xi_i = 1}
= beta$ (for an arbitrary $0 < be < 1$).
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